Sensitivity Analyses

# Sensitivity Analyses

**Session 19**

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# Unmeasured confounder

.box-1.medium[Untestable (but necessary!) assumption]

.box-1.medium[What happens if it is wrong?]

.box-inv-1.medium[Bias!]

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# Meaningful confounders

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# All you need

.box-1.medium[Exposure-outcome effect]

.box-1.medium[Exposure-unmeasured confounder effect]

.box-1.medium[Outcome-unmeasured confounder effect]

---

# All you need

.box-inv-1.medium[Exposure-outcome effect]

.box-2.medium[Exposure-unmeasured confounder effect]

.box-2.medium[Outcome-unmeasured confounder effect]

---
class: title title-1

# Exposure-Outcome effect

]

# Exposure-Outcome effect

<br>

# Exposure-Outcome effect

<br>

<br>

<br>

<br>

<br>

<br>

# All you need

.box-2.medium[Exposure-outcome effect]

.box-inv-1.medium[Exposure-unmeasured confounder effect]

.box-inv-1.medium[Outcome-unmeasured confounder effect]

---

# All you need

.box-2.medium[Exposure-outcome effect]

.box-inv-1.medium[Exposure-unmeasured confounder effect]

<br>

.box-inv-1.medium[Outcome-unmeasured confounder effect]

---

# All you need

.box-2.medium[Exposure-outcome effect]

.box-inv-1.medium[Exposure-unmeasured confounder effect]

.box-inv-1.medium[Outcome-unmeasured confounder effect]

<br>

---

# Sensitivity Analysis

.box-1[
`$$\beta_{Y\sim X |U +\mathbf{Z}} = \beta_{Y\sim X | \mathbf{Z}}-\beta_{Y\sim U|X+\mathbf{Z}}\times d$$`
]

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# tipr

```r
library(tipr)
adjust_coef(effect = 1.5,
            smd = 0.1,
            outcome_association = 1)
```

---

# tipr

```r
library(tipr)
*adjust_coef(effect = 1.5,
            smd = 0.1,
            outcome_association = 1)
```

---

# tipr

```r
library(tipr)
adjust_coef(effect = 1.5,
*           smd = 0.1,
            outcome_association = 1)
```

---

# tipr

```r
library(tipr)
adjust_coef(effect = 1.5,
            smd = 0.1,
*           outcome_association = 1)
```
---

# tipr

```r
library(tipr)
adjust_coef(effect = 1.5,
            smd = 0.1,
            outcome_association = 1)
```

```
## # A tibble: 1 × 4
##   effect_adjusted effect_observed   smd outcome_association
##             <dbl>           <dbl> <dbl>               <dbl>
## 1             1.4             1.5   0.1                   1
```

---

# tipr

```
The observed effect (1.5) is updated to 1.4 by a confounder with
the following specifications:
* estimated difference in scaled means: 0.1
* estimated association between the unmeasured confounder and 
*  the outcome: 1
```

---

# <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 640 512"><path d="M624 416H381.54c-.74 19.81-14.71 32-32.74 32H288c-18.69 0-33.02-17.47-32.77-32H16c-8.8 0-16 7.2-16 16v16c0 35.2 28.8 64 64 64h512c35.2 0 64-28.8 64-64v-16c0-8.8-7.2-16-16-16zM576 48c0-26.4-21.6-48-48-48H112C85.6 0 64 21.6 64 48v336h512V48zm-64 272H128V64h384v256z"/></svg> Application Exercise

1. `devtools::install_github("LucyMcGowan/tipr")`
2. You observed a causal effect of 2 (coefficient from a linear model). You think there could be an unmeasured confounder with a difference in scaled means of 0.5 and a outcome association of 1.2. Use the `adjust_coef()` function to update your observed effect
3. Try to find an unmeasured confounder that would *nullify* your result (make it 0).
4. Knit, commit, push to GitHub

---
class: title title-1

# All you need

.box-2.medium[Exposure-outcome effect]

.box-inv-1.medium[Exposure-unmeasured confounder effect]

.box-2.medium[Outcome-unmeasured confounder effect]

---

# All you need

.box-2.medium[Exposure-outcome effect]

.box-2.medium[Exposure-unmeasured confounder effect]

.box-inv-1.medium[Outcome-unmeasured confounder effect]

---

# Tipping Point Analysis

.box-1.medium[What will tip our confidence bound to cross the null?]

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# Tipping Point Analysis

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# Tipping Point Analysis

.box-1[
`$$\beta_{Y\sim U |X+\mathbf{Z}, \textrm{tip} = 0} =\frac{\beta_{Y\sim X | \mathbf{Z}}}{d}$$`
]

---

# tipr

```r
library(tipr)
tip_lm(effect = 1.5,
        smd = 0.1)
```

```
## # A tibble: 1 × 4
##   observed_effect   smd outcome_association n_unmeasured_confounders
##             <dbl> <dbl>               <dbl>                    <dbl>
## 1             1.5   0.1                  15                        1
```

---

# tipr

```
The observed effect (1.5) WOULD be tipped by 1 unmeasured 
confounder with the following specifications:
  * estimated difference in scaled means between the 
    unmeasured confounder in the exposed population and 
    unexposed population: 0.1
  * estimated association between the unmeasured confounder 
    and the outcome: 15
```

---

# tipr

```r
library(tipr)
tip_lm(effect = 1.5,
        outcome_association = 1)
```

```
## # A tibble: 1 × 4
##   observed_effect   smd outcome_association n_unmeasured_confounders
##             <dbl> <dbl>               <dbl>                    <dbl>
## 1             1.5   1.5                   1                        1
```

---

# tipr

```
The observed effect (1.5) WOULD be tipped by 1 unmeasured 
confounder with the following specifications:
  * estimated difference in scaled means between the 
    unmeasured confounder in the exposed population 
    and unexposed population: 1.5
  * estimated association between the unmeasured confounder 
    and the outcome: 1
```

---

1. You observed a causal effect of 2 (coefficient from a linear model). You think there could be an unmeasured confounder with a difference in scaled means of 0.5. Use the `tip_lm()` function to find an unmeasured confounder-outcome relationship that would be needed to render your causal effect inconclusive
2. Knit, commit, push to GitHub

---

# Sensitivity analysis

.box-1[Under the assumption that the sensitivity parameters are
fixed, the variance of the observed effect is the same as the variance of the adjusted effect]

.box-1[This allows all adjustments to apply to confidence intervals the same way they would apply to point estimates]

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<img src="img/table-sens.png" width = "70%"></img>
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class: center, middle

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1. Try applying these techniques to your Lab 04